Nuprl Lemma : pv11_p1_inv_acc
∀Cmd:ValueAllType. ∀ldrs_uid:Id ⟶ ℤ. ∀mf:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(mf)). ∀e1:E.
∀z:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List).
(z ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;mf)(e1)
⇒ let ballot_num,accepted = z
in (∀bnum:pv11_p1_Ballot_Num(). ∀p:ℤ × Cmd.
((<bnum, p> ∈ accepted) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) bnum ballot_num))))
∧ l-ordered(pv11_p1_Ballot_Num() × ℤ × Cmd;pv1,pv2.↑(pv11_p1_leq_bnum(ldrs_uid) (fst(pv1)) (fst(pv2)));accepted)
∧ no_repeats(pv11_p1_Ballot_Num() × ℤ × Cmd;accepted))
Proof
Definitions occuring in Statement :
pv11_p1_AcceptorState: pv11_p1_AcceptorState(Cmd;ldrs_uid;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_leq_bnum: pv11_p1_leq_bnum(ldrs_uid),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
vatype: ValueAllType,
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
l-ordered: l-ordered(T;x,y.R[x; y];L)
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
l_all: (∀x∈L.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
listp: A List+,
name: Name,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
length: ||as||,
list_ind: list_ind,
pv11_p1_headers: pv11_p1_headers(),
cons: [a / b],
nil: [],
it: ⋅,
true: True,
select: L[n],
subtract: n - m,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
null: null(as),
pv11_p1_AcceptorState: pv11_p1_AcceptorState(Cmd;ldrs_uid;mf),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
State2: State2(init;tr1;X1;tr2;X2),
pi1: fst(t),
pv11_p1_p1a'base: pv11_p1_p1a'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
uiff: uiff(P;Q),
bnot: ¬bb,
assert: ↑b,
sq_stable: SqStable(P),
pv11_p1_init_acceptor: pv11_p1_init_acceptor(Cmd),
pi2: snd(t),
pv11_p1_init_accepted: pv11_p1_init_accepted(Cmd),
top: Top,
pv11_p1_on_p1a: pv11_p1_on_p1a(Cmd;ldrs_uid),
pv11_p1_leq_bnum: pv11_p1_leq_bnum(ldrs_uid),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
pv11_p1_max_bnum: pv11_p1_max_bnum(ldrs_uid),
bool: 𝔹,
unit: Unit,
pv11_p1_leq_bnum': pv11_p1_leq_bnum'(ldrs_uid),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
sq_type: SQType(T),
pv11_p1_dummy_ballot: pv11_p1_dummy_ballot(),
exists: ∃x:A. B[x],
cand: A c∧ B,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
pv11_p1_on_p2a: pv11_p1_on_p2a(Cmd;ldrs_uid),
let: let,
pv11_p1_add_if_new: pv11_p1_add_if_new(),
pv11_p1_same_pvalue: pv11_p1_same_pvalue(Cmd),
bor: p ∨bq,
lt_int: i <z j,
Id: Id,
outl: outl(x),
isl: isl(x),
exposed-it: exposed-it,
pv11_p1_same_proposal: pv11_p1_same_proposal(Cmd)
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}mf:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}e\000Cs:EO+(Message(mf)).
\mforall{}e1:E. \mforall{}z:pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List).
(z \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;mf)(e1)
{}\mRightarrow{} let ballot$_{num}$,accepted = z
in (\mforall{}bnum:pv11\_p1\_Ballot\_Num(). \mforall{}p:\mBbbZ{} \mtimes{} Cmd.
((<bnum, p> \mmember{} accepted) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) bnum bal\000Clot$_{num}$))))
\mwedge{} l-ordered(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd;pv1,pv2.\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid\000C}$) (fst(pv1))
(fst(pv2)));accepted)
\mwedge{} no\_repeats(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd;accepted))
Date html generated:
2016_05_17-PM-02_52_02
Last ObjectModification:
2016_01_18-AM-11_28_06
Theory : paxos!synod
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